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Example:
Consider the following numerical example, where we show
f
(
x, y
)
in a table.
x
f
(
x, y
)
012
1
.1
.2
.3
y
2
.2
.1
.1
for example
f
(0
,
2) =
P
(
X
=0
and
Y
=2
)=0
.
2
.
We can check that
f
(
x, y
)
is a proper
joint probability function since
f
(
x, y
)
≥
0
for all 6 combinations of
(
x, y
)
and the sum of these 6
probabilities is 1. When there are only a few values for
X
and
Y
it is often easier to tabulate
f
(
x, y
)
than to ﬁnd a formula for it. We’ll use this example below to illustrate other deﬁnitions for multivariate
distributions, but ﬁrstwegiveashortexamplewhereweneedto ﬁnd
f
(
x, y
)
.
Example:
Suppose a fair coin is tossed 3 times. Deﬁne the r.v.’s
X
= number of Heads and
Y
=1(0)
if
H
(
T
)
occurs on the ﬁrst toss. Find the joint probability function for
(
X,Y
)
.
Solution:
First we should note the range for
(
)
, which is the set of possible values
(
x, y
)
which
can occur. Clearly
X
canbe0
,1
,2
,or3and
Y
can be 0 or 1, but we’ll see that not all 8 combinations
(
x, y
)
are possible.
We can ﬁnd
f
(
x, y
)=
P
(
X
=
x, Y
=
y
)
by just writing down the sample space
S
=
{
HHH,HHT,HTH,THH,HTT,THT,TTH,TTT
}
that we have used before for this process.
Then simple counting gives
f
(
x, y
)
as shown in the following table:
x
f
(
x, y
)
0123
0
1
8
2
8
1
8
0
y
1
0
1
8
2
8
1
8
For example,
(
)=(0
,
0)
if and only if the outcome is
TTT
;(
)=(1
,
0)
iff the outcome
is either
THT
or
TTH
.